The matrix-valued hypergeometric equation

Abstract

The hypergeometric differential equation was found by Euler [Euler, L. (1769) Opera Omnia Ser. 1, 11–13] and was extensively studied by Gauss [Gauss, C. F. (1812) Comm. Soc. Reg. Sci. II 3, 123–162], Kummer [Kummer, E. J. (1836) Riene Ang. Math. 15, 39–83; Kummer, E. J. (1836) Riene Ang. Math. 15, 127–172], and Riemann [Riemann, B. (1857) K. Gess. Wiss. 7, 1–24]. The hypergeometric function known also as Gauss' function is the unique solution of the hypergeometric equation analytic at z = 0 and with value 1 at z = 0. This function, because of its remarkable properties, has been used for centuries in the whole subject of special functions. In this article we give a matrix-valued analog of the hypergeometric differential equation and of Gauss' function. One can only speculate that many of the connections that made Gauss' function a vital part of mathematics at the end of the 20th century will be shared by its matrix-valued version, discussed here.

The hypergeometric equation is a second-order differential equation with three regular singular points. This equation was found by Euler (1) and was studied extensively by Gauss (2), Kummer (3, 4), and Riemann (5). Using a linear fractional transformation, we can place the three singularities at 0, 1, and ∞. Accordingly, the equation becomes

[1]

This is Euler's hypergeometric differential equation. The hypergeometric function known also as Gauss' function is defined by the hypergeometric series

for |z| < 1 and by analytic continuation elsewhere. If c is not an integer, then ₂F₁(c, a, b; z) is the only solution of Eq. 1 analytic at z = 0 and with value 1 at z = 0. See ref. 6 for a complete account on the subject.

This function, because of its remarkable properties, has been used for centuries in the whole subject of special functions. In the past 30 years, the discoveries of new special functions and of applications of special functions to new areas of mathematics have initiated a resurgence of interest in this field.

Representation theory provides a very important approach to the study of special functions. Historically this approach dates to the classical papers of Cartan (7) and Weyl (8) in which they showed that spherical harmonics arise in a natural way from a study of functions on G/K, where G is the orthogonal group in n space and K denotes the orthogonal group in (n- 1) space. To get a theory applying to larger classes of special functions, it is necessary to drop the assumption that G is compact and also to consider functions not just on G/K but also on G with values in End(V), where V is any finite dimensional complex vector space. Let K̂ denote the set of all equivalence classes of complex finite dimensional irreducible representations of K; for each δ ∈ K̂, let ξ_δ denote the character of δ, d(δ) the degree of δ, and χ_δ = d(δ)ξ_δ. A spherical function Φ on G (see ref. 9) of type δ ∈ K̂ is a continuous function on G with values in End(V) such that Φ(e) = 1 and

The first general results were obtained by Gelfand in 1950 (10), who considered spherical functions of trivial type for Riemannian symmetric pairs (G, K); a short time thereafter the fundamental papers of Godement (11) and Harish-Chandra (12, 13) appeared. It turns out that the spherical functions of trivial type for a rank-one Riemannian symmetric pair, when G is suitably parametrized, can be identified with hypergeometric functions. In ref. 14 one finds a detailed elaboration of this theory for any K type when the symmetric space G/K is the complex projective plane.

One can only speculate that many of the connections that made Gauss' function a vital part of mathematics at the end of the 20th century will be shared by its End(V) valued version discussed here. It is natural to wonder whether the spherical functions of any type, associated to a rank-one Riemannian symmetric pair, can be expressed in terms of these matrix-valued hypergeometric functions and to study their relation with the relatively new theory of matrix-valued orthogonal polynomials.

There are two other important programs to generalize the classical hypergeometric equation. One is due to Gelfand (15) and his school and the other to Gross (16). In the first, the generalization involves scalar valued functions of several variables, whereas in the second, one is dealing with scalar valued functions of a matrix argument.

Moreover, it is worth observing that the abstract hypergeometric equation considered by Hille (17),

can be written in the form of Eq. 2.

Let V be a d-dimensional complex vector space. Given A, B, C ∈ End(V), let us consider the following hypergeometric equation

[2]

where F denotes a function on ℂ with values in V.

Let us look for solutions of the form F = z^αG, with G analytic at z = 0 and G(0) ≠ 0. We have

From Eq. 2 the following differential equation for G follows,

If we put G(z) = ∑_n≥0 zⁿG_n, we obtain the following recursion relation for the coefficients G_n. For all k ≥ -2 we have

For k = -2 we have α(C + α- 1)G₀ = 0 from which we derive the following indicial equation:

[3]

Let β₁,..., β_d be the eigenvalues of C; then the roots of the indicial equation are α = 0, 1- β₁,..., 1- β_d.

For α = 0 we obtain the solutions F of Eq. 2, analytic at z = 0, the Taylor coefficients of which are given by the recursion relation

[4]

If the eigenvalues of C are not 0, -1, -2,..., then the function F is characterized by its value at 0, because the matrix (k + 2)(C + k + 1) is nonsingular for all k ≥ -1.

Let us introduce the notation

for all m ≥ 0 and (C, A, B)₀ = 1. We observe that

Theorem 1. If {F_n}_n≥0 is a sequence in V that satisfies the recursion relation (Eq. 4) and , then

1. F_n+1 = [1/(n + 1)](C + n)^-1(A + n)(B + n)F_n, for all n ≥ 0, and

2. F_n = (1/n!)(C, A, B)_nF₀, for all n ≥ 0.

Proof: For (i), if we put k = -1 in Eq. 4 we get F₁ = C^-1ABF₀. Now let n ≥ 1 and set k = n- 1 in Eq. 4; then

For k = n- 2 we obtain

Therefore,

which proves (i). The statement in (ii) follows directly from (i) by induction on n ≥ 0.

Definition 1: If A, B, C ∈ End(V) and no eigenvalues of C are in the set {0, -1, -2,...}, we define

In the next theorem we summarize our results on the analytic solutions at z = 0 of differential Eq. 2.

Theorem 2. If then

1. the function ia analytic on |z| < 1 with values in End(V), and

2. if F₀ ∈ V, then F(z) = ₂F₁(^{A, B} C; z)F₀ is a solution of the hypergeometric equation

   

   such that F(0) = F₀. Conversely any solution F analytic at z = 0 is of this form.

Let β ≠ 1 be an eigenvalue of C. Then α = 1- β is a root of indicial Eq. 3. Now we want to study a sequence {G_k}_k≥0 satisfying

[5]

For k = -2, α(C + α- 1)G₀ = 0 if and only if G₀ is an eigenvector of C of eigenvalue β.

If , then {G_k}_k≥0 is determined by G₀.

Theorem 3. If {G_n}_n≥0 is a sequence in V that satisfies the recursion relation (Eq. 5), and G₀ ∈ V is an eigenvector of C of eigenvalue β, then

1. G_n+1 = [1/(α + n + 1)](C + α + n)^-1(A + α + n) × (B + α + n)G_n, for all n ≥ 0, and

2. G_n = [1/(α + 1)_n](C + α, A + α, B + α)_nG₀, for all n ≥ 0.

Proof: For (i), if we put k = -1 in Eq. 5 we obtain

which proves (i) for n = 0. Now for n ≥ 1 the proof continues along the same line as in Theorem 2. The statement in (ii) follows directly from (i) by induction on n ≥ 0.

Definition 2: If A, B, C ∈ End(V) and -α ∉ , then we define the function

Notice that for α = 0 we have

Theorem 4. If , then is analytic on |z| < 1 with values in End(V).

If is an eigenvalue of C and G₀ ∈ V is an eigenvector of C of eigenvalue β, then

is a solution of the hypergeometric equation

Observe that when V = ℂ and C = c, A = a, B = b are complex numbers, then β = c and the above function coincides with z^1-c₂F₁(^{a-c+1, b-c+1} 2-c; z) for G₀ = 1.

Corollary 1. Let C be diagonalizable and let V(β_i) be the eigenspace of C of eigenvalue β_i. Let be a basis of V and let {G_i,j}_j be a basis of V(β_i) for each β_i. If and O is a simply connected region in with 0 ∈ O, then

is a basis of the space of all solutions of the hypergeometric equation

analytic at z = 0. Moreover,

is a basis of the space of all analytic solutions on O.

When V = ℂ, a differential equation of the form

[6]

with u, v, c ∈ ℂ, after solving a quadratic equation, becomes

[7]

This is not necessarily the case when dim(V) > 1. In other words, a differential equation of the form

[8]

with U, V, C ∈ End(V), cannot always be reduced to one of the form of Eq. 2, because a quadratic equation in a noncommutative setting as End(V) may have no solutions. Thus, it is important to notice how to obtain the solutions of Eq. 8.

If the eigenvalues of C are not 0, -1, -2,... let us introduce the sequence [C, U, V]_n ∈ End(V) by defining inductively

for all n ≥ 0.

Definition 3: If U, V, C ∈ End(V) and no eigenvalues of C are in the set {0, -1, -2,...}, we define

If , then we define the function

Notice that for α = 0 we have

and that if U = 1 + A + B, V = AB, then

Now Theorem 4 and Corollary 1 generalize, mutatis mutandis, in the following way.

Theorem 5.

1. If , the function is analytic on |z| < 1 with values in End(V).

2. If is an eigenvalue of C and G₀ ∈ V is an eigenvector of C of eigenvalue β, then

   

   is a solution of the differential equation

   

Corollary 2. Let C be diagonalizable, and let V(β_i) be the eigenspace of C of eigenvalue β_i. Let be a basis of V, and let {G_i,j}_j be a basis of V(β_i) for each β_i. If and O is a simply connected region in with 0 ∈ Ō, then

is a basis of the space of all solutions of the differential equation

analytic at z = 0. Moreover,

is a basis of the space of all analytic solutions on O.

Example: Following Grünbaum (see ref. 18), let us consider the differential equation

[9]

where F denotes a function on ℂ with values in M(2, ℂ), and X, U, V, and W are the matrices

where α, β ∈ ℂ and j = 0, 1,....

The term FW in Eq. 9 forces us to consider this equation as a differential equation on functions that take values on ℂ⁴ and to consider the left and right multiplication by matrices in as linear maps in ℂ⁴. Thus, instead of Eq. 9, we shall consider the following equivalent differential equation

[10]

where

where I in the matrices above denotes the 2 × 2 identity matrix, and

It is easy to verify that the matrices A and B given below satisfy Ũ = 1 + A + B, T̃ = AB.

The parameters x₁, x₃, y₁, and y₃ are only subject to the following conditions:

It is important to notice that spec(C) = {α + 1, α + 2}, where each eigenvalue has multiplicity 2. We also remark that (A + k)(B + k) is, generically, nonsingular for k ≠ j, and that the kernel of (A + j)(B + j) is two-dimensional. Thus, is not a polynomial function, as in the classical case, but nevertheless we have the following result.

Corollary 3. Differential Eq. 9 is equivalent to a hypergeometric equation of the form of Eq. 2. Therefore, the Jacobi polynomials introduced by Grünbaum are given in terms of hypergeometric functions, namely

This gives an explicit example within the theory of matrix-valued orthogonal polynomials initiated by Krein (19).